# Questions tagged [uniform-spaces]

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49
questions

**10**

votes

**2**answers

857 views

### Uniform spaces as condensed sets

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Unif{Unif}\DeclareMathOperator\CHaus{CHaus}\DeclareMathOperator\Set{Set}\DeclareMathOperator\op{op}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\...

**0**

votes

**0**answers

34 views

### fundamental system of entourages of direct limit of uniform spaces

Suppose that we have a tower of uniform spaces $$X_1\subset X_2 \subset \ldots \subset X_n\subset \ldots$$ such that each embedding is uniformly continuous. Let $X=\bigcup_{n\in \mathbb N} X_n$, and ...

**0**

votes

**1**answer

79 views

### What is the definition of a prorelation?

In the context of quasi-uniform spaces, what is a prorelation?
In the text I'm reading, they're defined as a down-directed upper set on relations X->Y.
Now, I'm fine with a down-directed up-set, but ...

**9**

votes

**6**answers

566 views

### Open mapping theorem for complete non-metrizable spaces?

The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's ...

**3**

votes

**1**answer

180 views

### Does each $\omega$-narrow topological group have countable discrete cellularity?

A topological space $X$ is defined to have countable discrete cellularity if each discrete family of open subsets of $X$ is at most countable.
A family $\mathcal F$ of subsets of a topological space ...

**2**

votes

**1**answer

108 views

### Uniformly Converging Metrization of Uniform Structure

This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question.
Let $X$ be a set with a uniform structure ...

**35**

votes

**3**answers

2k views

### What is the structure preserved by strong equivalence of metrics?

Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...

**4**

votes

**1**answer

131 views

### What can say about $2^X= \{A\subseteq X: A\text{ is closed set} \}$, when $(X, \mathcal{U})$ is a compact uniform space?

It is known that if $(X, d)$ is a compact metric space, then hyperspace $2^X= \{A\subseteq X: A\text{ is closed set} \}$ is a compact space with Hausdorff metric
What can say about $2^X= \{A\...

**4**

votes

**1**answer

418 views

### Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?

Recently I came to know about Atsuji space from the paper. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have found ...

**3**

votes

**1**answer

303 views

### Quotient of compact metrizable space in Hausdorff space

Lets $X$ be a compact metrizable space and $f:X\to Y$ be a quotient map such that $Y$ equipped with the quotient topology is Hausdorff. Thus $Y$ is metrizable. Lets $\sim$ be an equivalence relation ...

**7**

votes

**1**answer

444 views

### Totally bounded spaces and axiom of choice

Wikipedia article on totally bounded spaces states "... the completion of a totally bounded space might not be compact in the absence of choice." Where is the axiom of choice used, and do you need it ...

**4**

votes

**1**answer

129 views

### When is the unitary dual of a lscs group uniformizable?

Let $G$ be a locally compact, second countable group. We equip the unitary dual $\widehat{G}$ with the Fell topology. I am looking for conditions which guarantee that the topological space $\widehat{G}...

**0**

votes

**1**answer

74 views

### Topology generated by complete and incomplete uniformities [closed]

Does there exist a topology which can be induced simultaneously by a complete and an incomplete uniformity?

**4**

votes

**1**answer

173 views

### Convergent net in a quasi-uniform space which is not Cauchy

The proof of the result that every convergent net in a uniform space is Cauchy, employs symmetry of the uniform space. A quasi-uniform space lacks that symmetry. Is it possible then to find a ...

**7**

votes

**0**answers

179 views

### Results that are easier in a metric space

Are there any significant results in the theory of metric spaces that (are considerably more difficult to reproduce/have not been reproduced) in the theory of uniform spaces?
In particular, I'm ...

**5**

votes

**0**answers

206 views

### Has the Roelcke completion of a topological group any reasonable algebraic structure?

It is well-known that each topological group $G$ carries (at least) four natural uniformities:
the left uniformity $\mathcal L$, generated by the base $\{\{(x,y)\in G\times G:y\in xU\}:U\in\mathcal ...

**2**

votes

**1**answer

93 views

### Cartesian powers of uniform spaces

In the nlab entry on uniform spaces they speak about an "inherited uniform structure on function spaces". Namely, if $X$ is a set and $(Y,\mathfrak{U})$ is a uniform space, then $Y^X$ can be equipped ...

**2**

votes

**1**answer

109 views

### Construct a specific base for Fine uniformities in the diagonal(Entourages) case

For every uniformizable space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity.
To construct Fine uniformities, Let ...

**2**

votes

**1**answer

102 views

### The separated uniform space associated with $(X,\mathfrak{U})$

If $\mathfrak{U}$ is a not necessarily separated uniform structure for some set $X$, then an equivalence relation $R$ can be introduced on $X$ by letting $x R y$ provided $(x,y)\in U$ for every $U\in \...

**5**

votes

**3**answers

152 views

### A Mackey-Ahrens theorem for uniform spaces?

Let $X$ be a uniform space and $F(X)$ the vector space of all uniformly continuous real-valued functions over $X$. It is possible to express every bounded uniform semimetric $d$ on $X$ as $d(x,y) = ...

**1**

vote

**0**answers

89 views

### In a topological group $G$ with its lower uniformity, if $G$ is locally totally bounded, is its completion locally compact?

There has been work done on groups whose lower uniformity (or Roelcke uniformity) is totally bounded, e.g. the orthogonal group on a Hilbert space. This condition is equivalent to saying the lower ...

**5**

votes

**0**answers

129 views

### Is a successor to a successor to the trivial group topology on an Abelian group, totally bounded?

Is there an example of an Abelian group $G$ and group topologies $\cal S$ and $\cal T$ on it such that $\cal S$ is an immediate successor to the trivial topology on $G$ (i.e there is no other group ...

**10**

votes

**2**answers

851 views

### Baire Category Theorem for complete uniform spaces

The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ...

**4**

votes

**1**answer

218 views

### In the category of uniform spaces, is the completion of a quotient map also a quotient map?

I asked this question about 2 months ago on math.stackexchange, but so far I received neither comments nor answers.
Let $X$ and $Y$ be two Hausdorff uniform spaces. A surjective uniformly continuous ...

**4**

votes

**1**answer

277 views

### The subbase theorem for total boundedness

The uniform space analogue of Alexander's subbase lemma on compact subbase is (As we know, Alexander subbase lemma can be used to prove Tychonoff's theorem) :
Let $(X,\mathcal{U})$ be a uniform ...

**3**

votes

**1**answer

140 views

### Totally bounded group topologies on $\Bbb Q$ with trivial intersection of two neighborhoods

Are there totally bounded group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Q$ such that for some open sets $A\in\mathcal S$ and $B\in \mathcal T$ we have $A\cap B=\{0\}$?

**11**

votes

**1**answer

233 views

### Duality between large and small scale structures

A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure $\mathcal{C}$ (defined by a ...

**4**

votes

**0**answers

239 views

### The Haar integral on uniform spaces

Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability.
As ...

**4**

votes

**1**answer

2k views

### How is the notion of a Lipschitz structure on a manifold defined?

According to wikipedia, there is such a definition. $\:$ The candidate that I can come up with is
"an equivalence class of metrics that induce the topology and make the space locally bi-Lipschitz
to ...

**5**

votes

**2**answers

180 views

### A theorem of Markov about completely regular spaces and topological groups

In Pontriaguin's classic book Grupos continuos (in English Continuous Groups), says that A. Markov proved that:
There are topological groups that are not normal.
Furthermore, he says it is deduced ...

**1**

vote

**1**answer

98 views

### Existence of a moderate uniform structure on $\Bbb R$

A moderate uniform structure $\mathcal U$ on $\Bbb R$ is one for which
$\forall U\in \mathcal U, \exists n\in \Bbb N,\quad U^n=\Bbb R^2$
but
$ \not\exists n\in \Bbb N,\forall U\in \mathcal U,\quad U^...

**0**

votes

**2**answers

199 views

### Is there a normal space that is not uniformly normal

Let $(X,\mathcal D)$ be a uniform space and $A,B\subseteq X$. Let's say $A$ is uniformly inside $B$ and write $A\le B$ iff there's some entourage $D$ for which
$$(\forall a\in A)(D[a]\subseteq B)$$
A ...

**1**

vote

**1**answer

135 views

### Normal Uniform Spaces and their function uniform spaces

Let $(X,\mathcal D)$ be a normal (diagonal) uniform space and $G$ be the set of all homeomorphisms $f:X\to X$. Let $\Delta$ be the uniformity on $X^X$ (inherited by $G$) by subbase
$$\Lambda =\{ \{(f,...

**2**

votes

**1**answer

342 views

### Extend Homeomorphism to Uniformly Continuous Function

I have a space $A$ which is homeomorphic to the open $n$-ball $B_n$.
I'm trying to build a CW-complex with it, so
I want a continuous function from the closed ball $\overline{B}_n$
to the closure $\...

**1**

vote

**1**answer

154 views

### Precompact reflection in diagonal uniform spaces

Each diagonal uniform space $(X,\mathcal D)$ can be derived from the covering uniform space $(X,\Sigma_{\mathcal D})$ and each covering uniform space $(X,\Sigma)$ can be derived from the diagonal ...

**1**

vote

**1**answer

115 views

### Reference: uniformity of pointwise convergence has no countable base

Does anyone have a reference for the fact that the uniformity of pointwise convergence on real functions of $[0,1]$ (that is, the uniformity generated by the sets $\lbrace (f,g) : |f(x) - g(x)| < \...

**3**

votes

**3**answers

382 views

### Complete uniform spaces require complete metrics?

Hey all,
It is well-known that any uniformity is generated by the family of pseudometrics which are uniformly continuous from the product uniformity to $\mathbb{R}$. Further, the uniformity is ...

**1**

vote

**2**answers

255 views

### Uniformities generated by metrics.

Any uniformity on a set $X$ is generated by a family of pseudometrics on $X$. So if $(X,\mathcal D)$ is a uniform space there's a set $P$ of pseudometrics on $X$ with
$$\mathcal D=\left< \bigcup_{...

**1**

vote

**1**answer

531 views

### Tensor product of C*-algebras of bounded, uniformly continuous functions on metric spaces

This is a follow up question to this one.
If $X$ is a metric space, denote by $C_u(X)$ the $C^\ast$-algebra of all bounded, uniformly continuous functions on $X$ (with the sup-norm).
Do we have $...

**2**

votes

**1**answer

1k views

### Extending uniformly continuous functions on subspaces to non-metrizable compactifications

I have a complete metric space $Y$, some non-metrizable(!) Hausdorff compactification $Z$ of it and a subspace $X \subset Y$.
Furthermore, I do have a uniformly continuous function $f$ on $X$. So ...

**3**

votes

**2**answers

211 views

### For any entourage $U,V$ there's an entourage $W$ such that $U\circ W\subseteq V\circ U$

Let $(X,\mathcal U)$ be a uniform space and let $U\in \mathcal U$. Is this statement true?
$$\forall V\in \mathcal U, \exists W\in \mathcal U, U\circ W\subseteq V\circ U$$
I think if the above ...

**2**

votes

**1**answer

507 views

### A uniformity with a countable base is a pseudometric uniformity.

I need a proof for this proposition:
If a uniformity $\mathfrak U$ on $X$ has a
countable fundamental system of
entourages, then it can be defined by
a pseudometric on $X$.
which is the ...

**2**

votes

**1**answer

233 views

### Does every proximal outer measure, measure all open sets?

Let $\: \langle X,\delta\rangle \: $ be a separated proximity space.
Let $\: \mu^* \: : \: 2^{X} \: \to \: [0,+\infty] \: $ be a proximal outer measure.
Let $U$ be an open subset of $X$.
Does it ...

**12**

votes

**2**answers

563 views

### Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group? [xpost from math.SE]

And what else can be said, if so?
(Original math.SE post)
In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. (...

**4**

votes

**3**answers

454 views

### Does every Lindelof uniform space have a Lindelof completion?

Recall Lindelof = every open cover has a countable subcover. Is the question of the title answered by known material from some standard text on uniform spaces?
Note it is well known to be true for ...

**4**

votes

**1**answer

267 views

### Does the weak approximation theorem hold for general topological fields?

The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|_1,\ldots,|\cdot|_n,$ and letting $F_i$ denote $F$ with the topology from $|\cdot|_i$, ...

**7**

votes

**4**answers

2k views

### Finite dimensional vector spaces over a complete but not-necessarily-valued field

I'm essentially reopening this old question of Ricky Demer which was never fully answered.
Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and ...

**3**

votes

**1**answer

360 views

### Chaos in uniform spaces

Let $Dom$ be a uniform space, and $\hspace{.04 in}f$ be a continuous function from $Dom$ to itself satisfying:
For all non-empty open subsets $U$ and $V$ of $Dom$, there exists a natural
number $n$ ...

**14**

votes

**6**answers

3k views

### A good place to read about uniform spaces

I'd like to learn a bit about uniform spaces, why are they useful, how do they arise, what do they generalize, etc., without getting away from the context of general topology. I have to prepare an ...